Secondary critical exponent, large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values

In this paper, we consider the positive solution of the Cauchy problem for the following doubly degenerate parabolic equation

On the time continuity of entropy solutions

We show that any entropy solution u of a convection diffusion equation ?_t u + div F(u)-Df(u) = b{\partial_t u + {\rm div} F(u)-\Delta\phi(u) =b} in Ω × (0, T) belongs to C([0,T),L¹_loc(W)){C([0,T),L^1_{\rm loc}({\Omega}))} . The proof does not use the uniqueness of the solution.     

Solutions for Discrete Periodic Schr?dinger Equations with Spectrum 0

In this paper we study the discrete nonlinear equation

On the Euler–Lagrange equation for a variational problem: the general case II

In this paper we study the existence of a solution in ${L^\infty_{\rm loc}(\Omega)}In this paper we study the existence of a solution in L^￥_loc(W){L^\infty_{\rm loc}(\Omega)} to the Euler–Lagrange equation for the variational problem

inf[`(u)] + W1,￥0(W) òW (ID(?u) + g(u)) dx,                   (0.1)\inf_{\bar u + W^{1,\infty}_0(\Omega)} \int\limits_{\Omega} ({\bf I}_D(\nabla u) + g(u)) dx,\quad \quad \quad \quad \quad(0.1)      5. Asymptotically autonomous parabolic evolution equations      Roland Schnaubelt 《Journal of Evolution Equations》2001,1(1):19-37 We study the long-term behaviour of the parabolic evolution equation $\[u'(t)=A(t)u(t)+f(t), t>s,\quad u(s)=x. \]$\[u'(t)=A(t)u(t)+f(t), t>s,\quad u(s)=x. \] If A(t) A(t) converges to a sectorial operator A with s(A)?i \Bbb R = ? \sigma(A)\cap i \Bbb R =\emptyset as t?￥ t\to\infty , then the evolution family solving the homogeneous problem has exponential dichotomy. If also f(t)? f￥ f(t)\to f_\infty , then the solution u converges to the 'stationary solution at infinity', i.e., limt?￥u(t) = -A\sp-1f￥=:u￥,        limt?￥u￠(t)=0,        limt?￥A(t)u(t)=Au￥. \lim_{t\to\infty}u(t)= -A\sp{-1}f_\infty=:u_\infty, \qquad \lim_{t\to\infty}u'(t)=0, \qquad \lim_{t\to\infty}A(t)u(t)=Au_\infty. .      6. Transition-layer solutions of quasilinear elliptic boundary blow-up problems and dirichlet problems      Zong Ming Guo  Yao Yong Yan 《数学学报(英文版)》2011,27(11):2177-2190 We study profiles of positive solutions for quasilinear elliptic boundary blow-up problems and Dirichlet problems with the same equation: - eDp u = f(x,u)inW, - \varepsilon \Delta _p u = f(x,u)in\Omega ,      7. Nontangential Limits and Fatou-Type Theorems on Post-Critically Finite Self-Similar Sets      Ricardo A. Sáenz 《Journal of Fourier Analysis and Applications》2012,18(2):240-265 In this paper we study the boundary limit properties of harmonic functions on ℝ+×K, the solutions u(t,x) to the Poisson equation \frac?2 u?t2 + Du = 0,\frac{\partial^2 u}{\partial t^2} + \Delta u = 0,      8. Highly time-oscillating solutions for very fast diffusion equations      Juan Luis V��zquez  Michael Winkler 《Journal of Evolution Equations》2011,11(3):725-742 This work is concerned with the fast diffusion equation ut = ?·(um-1 ?u)        (*) u_t = \nabla \cdot \big(u^{m-1} \nabla u\big) \qquad (\star)      9. An L p -theory for stochastic integral equations      Wolfgang Desch  Stig-Olof Londen 《Journal of Evolution Equations》2011,11(2):287-317 We investigate the stochastic parabolic integral equation of convolution type u=k1*Apu +?k=1￥ k2*gk+u0,   t 3 0, u=k_1\ast A_pu +\sum\limits_{k=1}^{\infty} k_2\star g^k+u_0, \; t\geq 0,      10. Removability of an isolated singularity of solutions of the Neumann problem for quasilinear parabolic equations with absorption that admit double degeneration      O. M. Boldovskaya 《Ukrainian Mathematical Journal》2010,62(7):1040-1060 We consider the Neumann initial boundary-value problem for the equation ut = \textdiv( um - 1| Du |l- 1Du ) - up {u_t} = {\text{div}}\left( {{u^{m - 1}}{{\left| {Du} \right|}^{\lambda - 1}}Du} \right) - {u^p}      11. Damped-Driven KdV and Effective Equations for Long-Time Behaviour of its Solutions      Sergei B. Kuksin 《Geometric And Functional Analysis》2010,20(6):1431-1463 For the damped-driven KdV equation $ \dot{u}-{\nu}u_{xx} + u_{xxx} - 6uu_{x} = \sqrt{\nu}\,\eta(t, x), x \in S^1, \int udx \equiv \int \eta dx \equiv 0, $ with 0 < ν ≤ 1 and smooth in x white in t random force η, we study the limiting long-time behaviour of the KdV integrals of motions (I 1, I 2, . . . ), evaluated along a solution u ν (t, x), as ν → 0. We prove that for ${0 \leq \tau := {\nu}t \lesssim 1}For the damped-driven KdV equation [(u)\dot]-nuxx + uxxx - 6uux = ?{n} h(t, x), x ? S1, òudx o òhdx o 0, \dot{u}-{\nu}u_{xx} + u_{xxx} - 6uu_{x} = \sqrt{\nu}\,\eta(t, x), x \in S^1, \int udx \equiv \int \eta dx \equiv 0,      12. On the Hölder constant for the second order derivatives of admissible solutions to m-Hessian equations      N. M. Ivochkina 《Journal of Mathematical Sciences》2010,170(4):496-509 We construct an a priori estimate of the seminorm á uxx ña, [`(W)] {\left\langle {{u_{xx}}} \right\rangle_{\alpha, \bar{\Omega }}} for solutions to the problem Fm[ u ] = f;    u |?W = F {F_m}\left[ u \right] = f;\quad \left. u \right|{_{\partial \Omega }} = \Phi      13. Energy decay rates for solutions of the wave equation with boundary damping and source term      Jong Yeoul Park  Tae Gab Ha  Yong Han Kang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,9(1):235-265 In this paper, we consider the wave equation u" - Du = |u|r uu' - \Delta u = |u|^\rho u      14. Singular limits for 2-dimensional elliptic problem involving exponential nonlinearity with nonlinear gradient term      Sami Baraket  Ines Ben Omrane  Taieb Ouni 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(1):59-78 Given a bounded open regular set W ì \mathbbR2{\Omega \subset \mathbb{R}^2} and x1, x2, ?, xm ? W{x_1, x_2, \ldots, x_m \in \Omega}, we give a sufficient condition for the problem -div(a(u)?u) = r2 f(u) -{\rm div}(a(u)\nabla u)= \rho^{2} f(u)      15. A note on uniqueness of entropy solutions to degenerate parabolic equations in {\mathbb{R}^N}      Boris Andreianov  Mohamed Maliki 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(1):109-118 We study the Cauchy problem in \mathbbRN{\mathbb{R}^N} for the parabolic equation ut+div F(u)=Dj(u),u_t+{\rm div}\,F(u)=\Delta\varphi(u),      16. Fowler transformation and radial solutions for quasilinear elliptic equations. Part 2: nonlinearities of mixed type      Matteo Franca 《Annali di Matematica Pura ed Applicata》2010,189(1):67-94 We discuss the existence and the asymptotic behavior of positive radial solutions for the following equation: Dp u(x)+f(u,|x|)=0, \Delta_p u({\bf x})+f(u,|{\bf x}|)=0,      17. On the global well-posedness of a class of Boussinesq�CNavier�CStokes systems      Changxing Miao  Liutang Xue 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(6):707-735 In this paper we consider the following 2D Boussinesq–Navier–Stokes systems lll?t u + u ·?u + ?p = - n|D|a u + qe2       ?t q+u·?q = - k|D|b q               div u = 0{\begin{array}{lll}\partial_t u + u \cdot \nabla u + \nabla p = - \nu |D|^\alpha u + \theta e_2\\ \quad\quad \partial_t \theta+u\cdot\nabla \theta = - \kappa|D|^\beta \theta \\ \quad\quad\quad\quad\quad{\rm div} u = 0\end{array}}      18. Quenching phenomena for a non-local diffusion equation with a singular absorption      Raúl Ferreira 《Israel Journal of Mathematics》2011,184(1):387-402 In this paper we study the quenching problem for the non-local diffusion equation ut(x,t) = òW J(x - y)u(y,t)dy + ò\mathbbRN\W J(x - y)dy - u(x,t) - lu - p(x,t) {u_t}(x,t) = \int\limits_\Omega {J(x - y)u(y,t)dy + \int\limits_{{\mathbb{R}^N}\backslash \Omega } {J(x - y)dy - u(x,t) - \lambda {u^{ - p}}(x,t)} }      19. General Fractional Calculus,Evolution Equations,and Renewal Processes      Anatoly N. Kochubei 《Integral Equations and Operator Theory》2011,71(4):583-600       20. Periodic Positive Solution to a Class of Singular Second-Order Ordinary Differential Equations      Qingliu Yao 《Acta Appl Math》2010,110(2):871-883 This paper studies the existence of a positive solution to the second-order periodic boundary value problem u￠￠(t)+l(t)u(t)=f(t,u(t)),    0 < t < 2p,  u(0)=u(2p), u￠(0)=u￠(2p),u^{\prime \prime }(t)+\lambda (t)u(t)=f\bigl(t,u(t)\bigr),\quad 0相似文献    设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Asymptotically autonomous parabolic evolution equations

We study the long-term behaviour of the parabolic evolution equation $\[u'(t)=A(t)u(t)+f(t), t>s,\quad u(s)=x. \]$\[u'(t)=A(t)u(t)+f(t), t>s,\quad u(s)=x. \] If A(t) A(t) converges to a sectorial operator A with s(A)?i \Bbb R = ? \sigma(A)\cap i \Bbb R =\emptyset as t?￥ t\to\infty , then the evolution family solving the homogeneous problem has exponential dichotomy. If also f(t)? f_￥ f(t)\to f_\infty , then the solution u converges to the 'stationary solution at infinity', i.e., lim_t?￥u(t) = -A\sp-1f_￥=:u_￥,        lim_t?￥u￠(t)=0,        lim_t?￥A(t)u(t)=Au_￥. \lim_{t\to\infty}u(t)= -A\sp{-1}f_\infty=:u_\infty, \qquad \lim_{t\to\infty}u'(t)=0, \qquad \lim_{t\to\infty}A(t)u(t)=Au_\infty. .     

Transition-layer solutions of quasilinear elliptic boundary blow-up problems and dirichlet problems

We study profiles of positive solutions for quasilinear elliptic boundary blow-up problems and Dirichlet problems with the same equation:

Nontangential Limits and Fatou-Type Theorems on Post-Critically Finite Self-Similar Sets

In this paper we study the boundary limit properties of harmonic functions on ℝ₊×K, the solutions u(t,x) to the Poisson equation

Highly time-oscillating solutions for very fast diffusion equations

This work is concerned with the fast diffusion equation

An L p -theory for stochastic integral equations

We investigate the stochastic parabolic integral equation of convolution type

Removability of an isolated singularity of solutions of the Neumann problem for quasilinear parabolic equations with absorption that admit double degeneration

We consider the Neumann initial boundary-value problem for the equation

Damped-Driven KdV and Effective Equations for Long-Time Behaviour of its Solutions

For the damped-driven KdV equation $ \dot{u}-{\nu}u_{xx} + u_{xxx} - 6uu_{x} = \sqrt{\nu}\,\eta(t, x), x \in S^1, \int udx \equiv \int \eta dx \equiv 0, $ with 0 < ν ≤ 1 and smooth in x white in t random force η, we study the limiting long-time behaviour of the KdV integrals of motions (I ₁, I ₂, . . . ), evaluated along a solution u ^ν (t, x), as ν → 0. We prove that for ${0 \leq \tau := {\nu}t \lesssim 1}For the damped-driven KdV equation

[(u)\dot]-nuxx + uxxx - 6uux = ?{n} h(t, x), x ? S1, òudx o òhdx o 0, \dot{u}-{\nu}u_{xx} + u_{xxx} - 6uu_{x} = \sqrt{\nu}\,\eta(t, x), x \in S^1, \int udx \equiv \int \eta dx \equiv 0,      12. On the Hölder constant for the second order derivatives of admissible solutions to m-Hessian equations      N. M. Ivochkina 《Journal of Mathematical Sciences》2010,170(4):496-509 We construct an a priori estimate of the seminorm á uxx ña, [`(W)] {\left\langle {{u_{xx}}} \right\rangle_{\alpha, \bar{\Omega }}} for solutions to the problem Fm[ u ] = f;    u |?W = F {F_m}\left[ u \right] = f;\quad \left. u \right|{_{\partial \Omega }} = \Phi      13. Energy decay rates for solutions of the wave equation with boundary damping and source term      Jong Yeoul Park  Tae Gab Ha  Yong Han Kang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,9(1):235-265 In this paper, we consider the wave equation u" - Du = |u|r uu' - \Delta u = |u|^\rho u      14. Singular limits for 2-dimensional elliptic problem involving exponential nonlinearity with nonlinear gradient term      Sami Baraket  Ines Ben Omrane  Taieb Ouni 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(1):59-78 Given a bounded open regular set W ì \mathbbR2{\Omega \subset \mathbb{R}^2} and x1, x2, ?, xm ? W{x_1, x_2, \ldots, x_m \in \Omega}, we give a sufficient condition for the problem -div(a(u)?u) = r2 f(u) -{\rm div}(a(u)\nabla u)= \rho^{2} f(u)      15. A note on uniqueness of entropy solutions to degenerate parabolic equations in {\mathbb{R}^N}      Boris Andreianov  Mohamed Maliki 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(1):109-118 We study the Cauchy problem in \mathbbRN{\mathbb{R}^N} for the parabolic equation ut+div F(u)=Dj(u),u_t+{\rm div}\,F(u)=\Delta\varphi(u),      16. Fowler transformation and radial solutions for quasilinear elliptic equations. Part 2: nonlinearities of mixed type      Matteo Franca 《Annali di Matematica Pura ed Applicata》2010,189(1):67-94 We discuss the existence and the asymptotic behavior of positive radial solutions for the following equation: Dp u(x)+f(u,|x|)=0, \Delta_p u({\bf x})+f(u,|{\bf x}|)=0,      17. On the global well-posedness of a class of Boussinesq�CNavier�CStokes systems      Changxing Miao  Liutang Xue 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(6):707-735 In this paper we consider the following 2D Boussinesq–Navier–Stokes systems lll?t u + u ·?u + ?p = - n|D|a u + qe2       ?t q+u·?q = - k|D|b q               div u = 0{\begin{array}{lll}\partial_t u + u \cdot \nabla u + \nabla p = - \nu |D|^\alpha u + \theta e_2\\ \quad\quad \partial_t \theta+u\cdot\nabla \theta = - \kappa|D|^\beta \theta \\ \quad\quad\quad\quad\quad{\rm div} u = 0\end{array}}      18. Quenching phenomena for a non-local diffusion equation with a singular absorption      Raúl Ferreira 《Israel Journal of Mathematics》2011,184(1):387-402 In this paper we study the quenching problem for the non-local diffusion equation ut(x,t) = òW J(x - y)u(y,t)dy + ò\mathbbRN\W J(x - y)dy - u(x,t) - lu - p(x,t) {u_t}(x,t) = \int\limits_\Omega {J(x - y)u(y,t)dy + \int\limits_{{\mathbb{R}^N}\backslash \Omega } {J(x - y)dy - u(x,t) - \lambda {u^{ - p}}(x,t)} }      19. General Fractional Calculus,Evolution Equations,and Renewal Processes      Anatoly N. Kochubei 《Integral Equations and Operator Theory》2011,71(4):583-600       20. Periodic Positive Solution to a Class of Singular Second-Order Ordinary Differential Equations      Qingliu Yao 《Acta Appl Math》2010,110(2):871-883 This paper studies the existence of a positive solution to the second-order periodic boundary value problem u￠￠(t)+l(t)u(t)=f(t,u(t)),    0 < t < 2p,  u(0)=u(2p), u￠(0)=u￠(2p),u^{\prime \prime }(t)+\lambda (t)u(t)=f\bigl(t,u(t)\bigr),\quad 0相似文献    设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

On the Hölder constant for the second order derivatives of admissible solutions to m-Hessian equations

We construct an a priori estimate of the seminorm á u_xx ñ_{a, [`(W)]} {\left\langle {{u_{xx}}} \right\rangle_{\alpha, \bar{\Omega }}} for solutions to the problem

Energy decay rates for solutions of the wave equation with boundary damping and source term

In this paper, we consider the wave equation

Singular limits for 2-dimensional elliptic problem involving exponential nonlinearity with nonlinear gradient term

Given a bounded open regular set W ì \mathbbR²{\Omega \subset \mathbb{R}^2} and x₁, x₂, ?, x_m ? W{x_1, x_2, \ldots, x_m \in \Omega}, we give a sufficient condition for the problem

A note on uniqueness of entropy solutions to degenerate parabolic equations in {\mathbb{R}^N}

We study the Cauchy problem in \mathbbR^N{\mathbb{R}^N} for the parabolic equation

Fowler transformation and radial solutions for quasilinear elliptic equations. Part 2: nonlinearities of mixed type

We discuss the existence and the asymptotic behavior of positive radial solutions for the following equation:

On the global well-posedness of a class of Boussinesq�CNavier�CStokes systems

In this paper we consider the following 2D Boussinesq–Navier–Stokes systems

Quenching phenomena for a non-local diffusion equation with a singular absorption

In this paper we study the quenching problem for the non-local diffusion equation

Periodic Positive Solution to a Class of Singular Second-Order Ordinary Differential Equations

This paper studies the existence of a positive solution to the second-order periodic boundary value problem